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\begin{document}

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\lhead{Haojiong Shangguan}
\chead{Answers}
\rhead{2021/6/9}

\section*{1.Nonhomogenous boundary condition}
\begin{equation*}
  u(x) = exp(sin(x)) \quad f(x) = exp(x)(sin(x) - cos^2(x))
\end{equation*}
The residual and the reduction rate of the residuals for each V-cycle
are in the folder output1(make run). As for the convergence rate, through my
output, when the number of grids increases, iterate the same number of
steps and the magnitude of the error drops the same, but I can't find
the bug.

\section*{2.Homogenous boundary condition}
\begin{equation*}
  u(x) = -x^2+x \quad f(x) = 2
\end{equation*}
The residual and the reduction rate of the residuals for each V-cycle
are in the folder output2(make run). As for the convergence rate, through my
output, when the number of grids increases, iterate the same number of
steps and the magnitude of the error drops the same, but I can't find
the bug.

\section*{3.Reduce the accuracy to $2.2\times10^{-16}$}
\begin{equation*}
  relative \ accuracy = \frac{\Vert error \Vert_{\infty}}{\Vert precise\ solution \Vert_{\infty}}
\end{equation*}
The double-precision floating-point number has 16 significant bits, so
if the accuracy is reduced to $2.2\times10^{-16}$, the ratio of error
maxnorm to precise solution norm will never less than it, so the loop
will not stop. 


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